The very reason the impeachment proviso was placed into the Constitution was to cover the notion of a President committing a crime, or abusing his power as the Executive. This was to ensure that he would not be able to with, juridical impunity, break the law as he pleased and claim his office shielded him from prosecution for illegal or treasonable acts.
So yes, the President can commit a crime, and the Founders made it clear he could be punished or removed from office for so doing. They wanted to avoid the example of monarchy, where a king was essentially exempt from any judicial action on the basis of his stature of so-called 'divine right'.
Post-Nixon, I don't know WTF this is even a question in the US any longer.